Local regularity, multifractal analysis and boundary behavior of harmonic functions Eugenia Malinnikova NTNU, NORWAY; visiting Purdue University, IN Bloomington, October 10, 2015 E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 1 / 239
Outline Local regularity E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 2 / 239
Outline Local regularity Multifractal analysis and harmonic extension E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 2 / 239
Outline Local regularity Multifractal analysis and harmonic extension Positive harmonic functions E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 2 / 239
Outline Local regularity Multifractal analysis and harmonic extension Positive harmonic functions Hausdorff measures of sets of extremal growth E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 2 / 239
Outline Local regularity Multifractal analysis and harmonic extension Positive harmonic functions Hausdorff measures of sets of extremal growth Oscillation integral and the law of the iterated logarithm E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 2 / 239
Local regularity Let f : R → R and α > 0, we say that f ∈ C α ( x 0 ) if there exists a polynomial P of degree less than α such that | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α , | x − x 0 | < 1 . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 3 / 239
Local regularity Let f : R → R and α > 0, we say that f ∈ C α ( x 0 ) if there exists a polynomial P of degree less than α such that | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α , | x − x 0 | < 1 . The local Hölder exponent is h f ( x 0 ) = sup { α : f ∈ C α ( x 0 ) } . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 3 / 239
Local regularity Let f : R → R and α > 0, we say that f ∈ C α ( x 0 ) if there exists a polynomial P of degree less than α such that | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α , | x − x 0 | < 1 . The local Hölder exponent is h f ( x 0 ) = sup { α : f ∈ C α ( x 0 ) } . EXAMPLE: R ( x ) = � ∞ n 2 sin π n 2 x , 1 1 Riemann function, non-differential at x �∈ Q (Hardy, Littlewood) E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 3 / 239
Local regularity Let f : R → R and α > 0, we say that f ∈ C α ( x 0 ) if there exists a polynomial P of degree less than α such that | f ( x ) − P ( x − x 0 ) | ≤ C | x − x 0 | α , | x − x 0 | < 1 . The local Hölder exponent is h f ( x 0 ) = sup { α : f ∈ C α ( x 0 ) } . EXAMPLE: R ( x ) = � ∞ n 2 sin π n 2 x , 1 1 Riemann function, non-differential at x �∈ Q (Hardy, Littlewood) Jaffard (1996) computed h R ( x ) explicitly, 1 / 2 ≤ h R ( x ) ≤ 3 / 2 depends on the rate of rational approximation. E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 3 / 239
Wavelet transform Local regularity can be measured by the decay of the wavelet transform W f ( a , b ) = 1 � f ( t ) ψ ( a − 1 ( t − b )) dt , a R where ψ is a "wavelet-function", ψ is smooth enough and � ψ ( t ) dt = 0 . Roughly speaking, f ∈ C α ( x 0 ) iff | W f ( a , b ) | ≤ Ca α ( 1 + a − 1 | b − x 0 | ) α . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 4 / 239
Spectrum of singularities Let E f ( β ) = { x ∈ R : h f ( x ) = β } d f ( β ) = dim H ( E f ( β )) , d f is called the spectrum of singularities (multifractal spectrum) of f . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 5 / 239
Spectrum of singularities Let E f ( β ) = { x ∈ R : h f ( x ) = β } d f ( β ) = dim H ( E f ( β )) , d f is called the spectrum of singularities (multifractal spectrum) of f . EXAMPLE: d R ( β ) =? E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 5 / 239
Local dimension of a measure Let µ be a positive measure on R m − 1 , we define the (lower) local dimension of µ at x 0 as log µ ( B ( r , x 0 )) h µ ( x 0 ) = lim inf . log r r → 0 + When m = 2 then h µ ( x 0 ) = h F ( x 0 ) , where F is the anti-derivative of µ . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 6 / 239
Local dimension of a measure Let µ be a positive measure on R m − 1 , we define the (lower) local dimension of µ at x 0 as log µ ( B ( r , x 0 )) h µ ( x 0 ) = lim inf . log r r → 0 + When m = 2 then h µ ( x 0 ) = h F ( x 0 ) , where F is the anti-derivative of µ . (almost) E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 6 / 239
Local dimension of a measure Let µ be a positive measure on R m − 1 , we define the (lower) local dimension of µ at x 0 as log µ ( B ( r , x 0 )) h µ ( x 0 ) = lim inf . log r r → 0 + When m = 2 then h µ ( x 0 ) = h F ( x 0 ) , where F is the anti-derivative of µ . (almost) We will instead work with the harmonic extension u = P ∗ µ , we define F γ ( u ) = { y ∈ R m − 1 : lim sup u ( y , t ) t γ > 0 } . t → 0 E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 6 / 239
Local dimension of a measure Let µ be a positive measure on R m − 1 , we define the (lower) local dimension of µ at x 0 as log µ ( B ( r , x 0 )) h µ ( x 0 ) = lim inf . log r r → 0 + When m = 2 then h µ ( x 0 ) = h F ( x 0 ) , where F is the anti-derivative of µ . (almost) We will instead work with the harmonic extension u = P ∗ µ , we define F γ ( u ) = { y ∈ R m − 1 : lim sup u ( y , t ) t γ > 0 } . t → 0 Exercise The following estimate holds dim H F γ ( u ) ≤ m − 1 − γ, and it is sharp. E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 6 / 239
Generalized local dimension Let v be increasing on [ 0 , 1 ) , λ ( t ) = t m − 1 v ( t ) be increasing and lim t → 0 t m − 1 v ( t ) = 0. E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 7 / 239
Generalized local dimension Let v be increasing on [ 0 , 1 ) , λ ( t ) = t m − 1 v ( t ) be increasing and lim t → 0 t m − 1 v ( t ) = 0. Theorem (K.S. Eikrem, M., 2012; F. Bayart, Y. Heurteaux, 2013)) (i) Let u be a positive harmonic function in R m + , we define u ( y , t ) F v ( u ) = { y ∈ R m − 1 : lim sup > 0 } . v ( t ) t → 0 + Then F v ( u ) is a countable union of sets of finite H λ -measure. E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 7 / 239
Generalized local dimension Let v be increasing on [ 0 , 1 ) , λ ( t ) = t m − 1 v ( t ) be increasing and lim t → 0 t m − 1 v ( t ) = 0. Theorem (K.S. Eikrem, M., 2012; F. Bayart, Y. Heurteaux, 2013)) (i) Let u be a positive harmonic function in R m + , we define u ( y , t ) F v ( u ) = { y ∈ R m − 1 : lim sup > 0 } . v ( t ) t → 0 + Then F v ( u ) is a countable union of sets of finite H λ -measure. (ii) There exists a positive function u such that u ( y , t ) ≤ v ( t ) and H λ ( E v ( u )) > 0 , where u ( y , t ) E v ( u ) = { y ∈ S : lim inf > 0 } . v ( t ) t → 0 + E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 7 / 239
Generalized local dimension Let v be increasing on [ 0 , 1 ) , λ ( t ) = t m − 1 v ( t ) be increasing and lim t → 0 t m − 1 v ( t ) = 0. Theorem (K.S. Eikrem, M., 2012; F. Bayart, Y. Heurteaux, 2013)) (i) Let u be a positive harmonic function in R m + , we define u ( y , t ) F v ( u ) = { y ∈ R m − 1 : lim sup > 0 } . v ( t ) t → 0 + Then F v ( u ) is a countable union of sets of finite H λ -measure. (ii) There exists a positive function u such that u ( y , t ) ≤ v ( t ) and H λ ( E v ( u )) > 0 , where u ( y , t ) E v ( u ) = { y ∈ S : lim inf > 0 } . v ( t ) t → 0 + For a typical (Baire category) positive measure the set of given growth has exactly this "Hausdorff dimension" . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 7 / 239
Classes of harmonic functions of controlled growth Let v ( t ) , t > 0 , be a positive increasing continuous function and assume that lim t → 0 + v ( t ) = + ∞ . We define k v = { u : R m + → R , ∆ u = 0 , u ( y , t ) ≤ Kv ( t ) } , and h v = { u : R m + → R , ∆ u = 0 , | u ( y , t ) | ≤ Kv ( | t | ) } . E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 8 / 239
Classes of harmonic functions of controlled growth Let v ( t ) , t > 0 , be a positive increasing continuous function and assume that lim t → 0 + v ( t ) = + ∞ . We define k v = { u : R m + → R , ∆ u = 0 , u ( y , t ) ≤ Kv ( t ) } , and h v = { u : R m + → R , ∆ u = 0 , | u ( y , t ) | ≤ Kv ( | t | ) } . Similar spaces can be considered in the unit disc (ball). For any v there exists u ∈ h v such that u ( ry ) → ∞ for a.e. y ∈ S (N. Lusin, I. Privalov; J.-P. Kahane, Y. Katsnelson). This behavior is very different of the one we have seen for positive harmonic functions. E. Malinnikova (NTNU) Boundary behavior of harmonic functions MWAA2015 8 / 239
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